Axioms from Interactions of Actors Theory Nick Green icon

Axioms from Interactions of Actors Theory Nick Green

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Axioms from Interactions of Actors Theory

Nick Green

Real Time Study Group

Department of Computer Science

University of Wales, Cardiff

Keywords Cybernetics, Self-organisation, Interaction, Actors, Evolution, Learning, Spin

Abstract While working with clients in the last years of his life Gordon Pask produced an axiomatic scheme for his Interactions of Actors Theory which is a development of his well known Conversation Theory. These axioms are interpretable as a general theory of self-organisation and are discussed as characteristic of field concurrence and as part of the second order cybernetics canon. An application to population density is reported supported by both kinematic and kinetic simulation. Implications for cardiovascular anti-coagulation therapy and planetary evolution are discussed.


Gordon Pask's Interactions of Actors Theory (IA) was developed from his earlier Conversation Theory (CT). The key ideas are presented as the writer encountered them. A cosmological picture emerges of elementary, interacting self-organising processes that evolve to produce life and concepts in our brains. Pask regards concepts, in general, as any persistent self-productive looping process in any state of matter. CT requires the concurrent existence of at least two distinct precursor concepts to generate a third. These concepts resonate like chemical tautomers making them analogical to each other. CT is transformed to IA by the taking of duals and interacting Actors are produced which support participants in conversations of bounded duration. The central object is the stable concept triple which takes the form of the Borromean link. The potential of this form as a concurrent computing element and a model of continuity is discussed. The self-organising forces exerted by the concept triple are the subject of the axioms which are discussed in detail.

Lastly serial and concurrent experimental work is reported which applies Pask's Last Theorem (PLT), a theorem about differences and forces. Experiment confirms the view that weak forces lead to dense forms of self-organisation in ekistics (the theory of settlements), cardiovascular anti-coagulation therapy and planetary evolution. It will be noted, for example, that diatomic hydrogen cannot form unless Pask imperative forces act on the atoms.

In his later years Pask had effectively shown IA to be a theory of Cybernetics hence confirming the von Foerster soubriquet for him "Mister Cybernetics", the "cybernetician's cybernetician" (von Forster, 1995).

In 1994 while consulting with clients1 Gordon Pask made a clear statement of axioms or properties governing his Conversation Theory, CT (Pask, 1976), the means by which concepts are shared, and its underlying support Interaction of Actors Theory, IA (Pask, 1993). As Bateson (Bateson, 1980) has it Nature and Mind are one. To Pask, also, Nature's tendency to self-organisation produces concepts (Pask, 1996) whether in a star or a brain. A formal statement of these axioms has never been made. By describing how IA developed and the names and properties of the axioms as discussed with Pask in due course a more formal paper and scheme might be produced.

^ P-Individuals and M-individuals

There are two parts to an interacting participant or Actor. First the P-individual which is a dynamic, productive and incidentally reproductive, adaptive, evolving and learning collection or entailment mesh of concepts. Dawkins' memes (Dawkins, 1976) are a restricted form of P- individual. Second the M-individual which is a mechanical or biological medium e.g. a computer, a brain or a star, which supports the P-Individual and the strains its concepts produce. One P-individual may inhabit many M-individuals and many P-individuals may interpret in a single M-individual (Pask, 1975). To Pask Nature's eternal interactions, with associated P-individuals interpreted in M-individuals, could be regarded as conversations between participants when constrained by beginnings and ends. The constraints on the second order paradigm of Cybernetics, where one participant may be a carbon-based life form designated observer and part of the system, are made plain with the axioms. These axioms apply to all participants whether observer or not.

Pask (Pask,1990) won an award from Old Dominion University, Virginia for his Process/Product complementarity principle. "Every process produces a product, every product is produced by a process" e.g. electromagnetic waves and photons or in the case of CT applied concepts and their descriptions.

This may be written Ap(Conz(T)) => Dz(T) where => means produces, z is a participant and T the current topic or concept from which all others can unfold as thought proceeds from within the domain or entailment mesh of the participants. So the Description, D, of Concept, T, is produced by the Application, Ap, operator on the Conversation, Con, operator on T. The dual character is denoted: <Ap(Conz(T)), Dz(T)>. Physically D is a hard carapace of repulsive force.

P and M-Individuals are also dual process/product pairs. Pask wrote this as:
denoting an Actor or participant to interaction or conversation. Ap, Con and D are operators in Pask's protolanguage Lp or protologic as he sometimes called it.


A concept is derived by the conjunction, ANDing, of at least two concurrent concepts. This satisfies the marked state requirement of Petri Information transfer. Dr Bernard Scott (personal communication) points out that this is equivalent to the “irreducible to binary form” relations emphasised by Peirce, Korzybsky & McCulloch.

This may be shown graphically as in Figure 1.

Here, for example, we might be speaking of Newton's Second Law2. T1 could stand for the concept of Force with T2 and T3 being mass and acceleration. To understand the modern theory of force more deeply we might prefer the topics or concepts to be assigned as sparticle, knot and string. Something more trivial seeming might be dog, lead and walk. Clearly in all these examples a third concept can be derived from any pair. There is a strict analogy between any concept pair, distinguished by differences and similarities defined by the third concept. Pask (Pask, 1976) called this "local cyclicity" and this can be demonstrated graphically as in Figure 2.

In the case of the two examples taken from classical and supersymmetry force theory we can demonstrate consistency and coherence with, for example, dimensional analysis or probing into the observations that led to the construction of the concepts. In the case of the dog, the walk and the lead a supporting hypothesis may also be constructed. Further concepts are entailed supporting the disambiguation of dog, walk and lead. One might show that leads are used to take dogs for walks rather than, say, connect them to an electricity supply. Supporting hypotheses of this kind are implicit in the scientific examples and can also be made explicit. The terms assigned to the topic numbers must, at least, be defined unambiguously for the appropriate context. Thence addition of concepts leads to potentially very large entailment meshes representing the concepts of one or many participants to a conversation delimited by beginnings and ends supported by an interaction which is eternal.

^ Interactions of Actors

It was from consideration of local cyclicity3 in Conversation Theory that Interaction of Actors Theory was derived. Pask summoned me to the Athenaeum, one of his clubs, to show me with great enthusiasm some large drawings he had made, charmingly indifferent to the club rule: no papers in the bar. Duality and the taking of duals is a widely used technique in simplifying, for example, graph theory and electrical circuit theory. Calculations can sometimes be simplified. In graph theory nodes can be interchanged with arcs and in circuit theory inductances with capacitances and current sources with voltage sources. Further we find a three pointed "star" of resistances is electrically equivalent to a triangle of resistances. Electrical engineers call this Star Delta duality4. Pask took the dual with the bidirectional arcs of local cyclicity and the topic or concept nodes. Thus in one elegant transformation Conversation Theory was made dynamic. The old derivations, the sticks of what he called his "stick and ball model" (as figures 1 and 2) became formally circular, closed loop processes. The topic or concept nodes became the intersections of the fields produced by the closed, looping process. He asserted forces acted between the circular processes and that they existed in stable triples. The Begins and Ends of the application of concepts in Conversation Theory had been replaced by eternal, evolving kinetic interactions between organisationally closed and informationally open concept

loops, toruses with carapaces that maintain a boundary, a distinction. An Actor is anything that acts as a result of an interaction or a transfer of meaningful information

The axioms or properties below apply to both Conversation and Interactions of Actors Theory with the exception that conversations of P-individuals have Begins and Ends and may be interrupted by other conversations and so be nested. Actors and their M-individuals interact eternally.

^ Concept resonance

The closed toroidal processes which comprise the concepts of P-individuals exist as stable triples in which any pair is analogous to the other and distinguished by the third. Any two concepts may generate the third because of their resonant similarities and differences.

The resonance produced by an incident field produces an output radiative field. Knot Theory was a matter of some concern to Pask. He coined the term "tapestry" making his entailment mesh structure of the concepts of a participant coherent with knot theory. Whilst loops are always permitted, indeed are the nature of concepts, the intersections of CT become crossings in IA where crossings up and down with loops define knots. The crossing up or down rule of the knots or links was not decided but a recursive and nested Borromean form seemed most likely. This seems coherent with the superstring theory interpretation of force. Here knots in strings produce the sparticles and thence bosons of current force theory e.g. Pierre van Baal and Andreas Wipf (van Baal & Wipf, 2001) who postulate Hopf linked strings (two intersecting loops) produce force. A suitably energetic hadron collider experiment e.g. Utpal Chattopadhyay and Pran Nath (Chattopadhyay & Nath, 2001) may be able to decide if the Hopf or Pask's Borromean model is most respectable. A smaller new experiment by Long et al (Long et al, 2003) using planar oscillators at 100m lengths has examined some of string theory predictions of the need for extra dimensions. In some cases this would result in deviations from the Newton's inverse square law at distances less than 1mm. None were found.

The twelve arcs of the Borromean Ring form and the equilateral prismatic tensegrity of Buckminster Fuller were under study at the time I joined Pask Associates in 1993. The tensegrity provided a tractable force model comprising three repulsions and nine attractions corresponding to the diagonal rods and tension strings. In due course these were configured as elements of a potential concurrent computer. Pask wanted a machine in which the three struts of the prismatic tensegrity were periodically excited with make and break circuits feeding solenoids. Tiny electric motors attached to each strut were more successful and less liable to collapse. The phasor sum of these vibrations produces a fourth frequency that is a mechanical four wave mixing analogue5 (Shih, 1987). We had no theory of desirable frequencies. A variety of resonant frequencies could be inferred but a full vibrational analysis with strain gauges and spectrograph was an obvious next step. Pask felt inspiration, it seemed to me, holding the machine as its three axis vibrated and, as usual, fell apart. He was clearly looking for unusual phenomenon.

With the help of the Royal Institution a higher precision device was made with six equal mass and 15 cm length mild steel rods. There was potential danger here without a safety cage and thankfully Pask appeared to lose interest in pursuing this having made his statement, as it were, of the nature of concept triples as computing elements. This device mimics the phase conjugate mirror. Two waves are mixed in a non-linear medium. Any input wave is amplified if the two "pumping" waves are sufficiently intense, phase conjugated and reflected back converging to the input source. Phase conjugate oscillator technology is crucial to some to make practical beam weapons. The fact that the incident wave is phase conjugated has lead to the radiated wave being described as "time reversed".

One can share in Pask's delight at interpreting in rough hardware his concept triples as a dynamic computing component where the memory requirement was formally embodied as a highly restricted method of travelling backwards in time. Further precision work and miniaturisation is required to create a practical device. Confirmation that output is phase conjugated with the input wave could be a first step in validating the prototype prismatic tensegrity computer. Estimating its capacity as a storage medium for analogue waveforms for given implementation technologies might be a next step.

^ Concept kinematics and kinetics

Concepts nest recursively in triples within each other. As drawn by Pask they form tori like wires in a multi-core electric cable6. Repulsive forces are exerted by the concepts generating a carapace or hard protective shell around them as in figure 6. These forces further distinguish the domain of Interactions of Actors Theory from Conversation Theory in that Pask asserted IA to be a kinetic theory and CT to be kinematic because of its implicit begins and ends. Forces cause thoughts to change and we have the Last Theorem, as Pask called it, which states

"Like concepts repel, unlike concepts attract".

Pask's Last Theorem (PLT)

This statement is intended to embody all forces: weak, strong, gravitational and electric or magnetic, which give rise to the self-organising character we see in Nature. Here '+' directs into the not void (or something) and '-' means orthogonal to and "deflects similar concepts". This deflective force is "bearing the clockwise or anticlockwise signature of the process which creates it" Pask (Pask, 1993, p. 44). In a most difficult part of the IA manuscript a prepositional operator mesh is introduced with ontologies and analogies. Ap becomes permissive application and & (or IM) imperative application. There is a discussion of unfoldment, Un, as mentation and action in this context. Un is identified with '-' and seeking out further concepts in its neighbourhood to apply. The '+' process may be clockwise or anticlockwise in orientation and may not always close or "eat its own tail" but may disappear into the void (Pask, 1993, p. 77). On page 78 he says "No aether, electromagnetic or not, is needed only orthogonality, as in the electrical and magnetic components of an electromagnetic field and, here, represented by '+' and '-'".

After stating PLT for the first time to me (Green, 2001) I asked Pask "What kind of force?" After a considered pause he said "Just a force".


The properties of the axioms can be investigated and appreciated as characteristic of interaction in field concurrent n body systems, like the Newtonian solar system or the electrically charged protons and electrons of an atom or molecule. The outcomes of interaction are interpreted in the participant Actors.

Interpretation of Pask's theory of force as producing self-organisation is ongoing. Some not so obvious considerations from the classical dynamics of forces should be pointed out. First stable closed elliptical orbits are rare in n body simulations with random initial conditions but equal masses. Choreographies, by contrast, are common when masses are equal e.g. Cris Moore Sante Fe7, Carlos Simó (~2000 but undated pre-print from Paris Observatory). These trajectories form braids and are stable in that a small perturbation produces a precession only. This may imply braids are frequent in liquids and gases between like polarised molecules. For serial computing simulation tools see Acheson (Acheson,1997). Asymmetric attractive, sticky, aggregative generation producing a non-uniform Power Law or Bakian 1/f distribution of masses (Bak, 1997) is probably necessary for the stable seeming elliptical orbits we all expect. Bak's condition of Self-Organised Criticality can be met with the non-linear aggregation of attractive uniform aggregates or Actors (see later for more, Witten and Sander, 1981). The mechanism of the stickiness (or valency) is worth further study and may be partly tractable under serial digital simulation. Poincaré proved circular orbits in n body systems have zero probability. Later Kolmogorov conjectured orbital toroids were feasible under perturbation in non-dissipative systems. The proofs due to Arnold and Moser in 1962 and 1963 produced the famous KAM Theorem8. Henceforth in this paper "circular" process implies a KAM compliant process.

Whilst the laws of electric or magnetic attraction and repulsion are familiar, recall again the closed curved Newtonian process. There is repulsion without electromagnetic law in the resistance of a gyroscope to a couple. Laithwaite's9 apparently notorious observation while sitting on a rotating office chair with a spinning bicycle wheel in his hand that rotating the seat in one direction made the wheel lighter and in the other made the wheel heavier should be recalled.

The behaviour of the asymmetric Tippe (sometimes spelt Tippy) Top10 should be considered. Here is counter example to whatever dynamical minimisation principles (e.g. of Fermat, Maupertius or Hamilton) is operating. The top turns up-side down at some critical frequency, reversing its direction of spin and achieving an equilibrium but raising its centre of gravity to do so. Conventional minimisation principles assert themselves as the top slows down past the critical frequency and it topples, reversing spin again, with the centre of gravity minimising its potential energy as intuition and simple Newtonian dynamics demands. A picture of Bohr demonstrating the inverting Tippe Top to Fermi11 is held by the American Institute of Physics.

Prigogine's Group12 believes non-linear interaction produces self-organisation in irreversible dissipative processes. Simultaneous three body collisions are under investigation and spin interaction in quantum systems, for example. For Pask a spin reversed concept was formally different and by PLT attractive. Stable triples exhibit residual clockwise/anticlockwise parity.

Lastly, it is worth noting a relatively new phenomenon discovered by David Acheson (Acheson, 1993) and discussed in Acheson (Acheson, 1997). The stiffening of a soft carapace, as IA might interpret it, has been demonstrated. Around 30 inches of ordinary household flexible plastic covered curtain wire stands up straight when vertical periodic excitations are applied to the base with a mechanical vibrator at a resonant frequency. Acheson calls this gravity defying effect13 "Not quite the Indian Rope Trick".

Interactions of Actors Axioms

  • Context

  • Perspective

  • Responsible

  • Respectable

  • Amity

  • Agreement

  • Agreement-to-disagree (ATD)

  • Purpose

  • Unity not uniformity

  • Faith

  • Beginnings and Ends (CT)

  • Eternally interacting (IA)

  • Similarity and Difference

  • Adaptation

  • Evolution

  • Generation

  • Kinetic (IA)

  • Kinematic (CT)

  • Conservation of Meaningful Information Transfer both Permissive (Ap) and Imperative Application (Im)

  • Informational openness and Organisational closure.

  • Void and Not-Void

Exactly how to treat these axioms is the subject of some discussion. Pask's form of words for each axiom is particularly useful in maximising applicability to the so-called soft sciences. The aim is to be able to make statements about interacting people of form equivalent in robustness to those of conventional physical science: age of the Earth, the water will take ten minutes to boil and so on. There is no implied order of priority in these axioms. They exist concurrently as restrictions on the behaviour of all participants in their interactions and the forces acting be they strings, sub-atomic particles, atoms, molecules, plasmas, gases, liquids, solids, plants or animals.

The setting up of counting and in particular state counting or variety may prove an interesting first challenge to elegance and chosen notation. The apparent simplicity of the Similarities and Differences in Actors Axiom are what may be deployed. It should be noted that a difference is the feedback of first order cybernetics. Beer once remarked to me that often people in conversation would make a statement or proposal and invite criticism or responses from others by saying "Give me some feedback". This he said was, with an emphatic pause, "not correct". Beer said no more and in the context of a formal model he may have a point. It would be correct in the context of the Difference axiom in CT/IA where "teach back", the feedback to which Beer objected, is judged, implied or used explicitly to confirm a new difference acquired or successful act of learning or meaningful information transfer. This is also described as the execution of a model in a modelling facility shared by participants, within the Cognitive Reflector form of CT, to demonstrate an interpretation of the newly acquired difference by a participant. This was Pask practising as philosophical mechanic (Pask, 1993, p. 83). These are more or less formal routine, but imperative, components of transactions in every day conversation. Cognitive pathologies result from their incorrect operation e.g. vacuous holism "Can't see the trees for the woods" or pathological serialism "Can't see the woods for the trees" rather than the desirable balance of local and global versatile cognitive style. The nonsense talked about micromanagement today reflects these pathologies. The dangerously fallacious (vacuous holist) belief that knowledge of detail prevents proper management lead to a Railtrack Board with no engineers14. The lack of proper performance of track maintenance sub-contractors seemed unnoticed by the Board. There were tragic consequences with seven dead outside London at Potters Bar on May 10th 2002. In 2003 investigative reporting in the London Evening Standard showed track maintenance sub-contractor costs out of control. BBC investigations disclosed a similar pattern at London Underground. A therapeutic dose of Viable System Theory can cure but participants get attached to their pathologies. They are surrounded with repulsive forces. They have to "want to change" as the psychotherapist might say. Presenting an unwelcome difference the professional cybernetician may be confronted with Dr Elizabeth Kübla-Ross' five stages of grief: "denial, anger, depression, bargaining and acceptance" (Kübla-Ross, 1969). Through enquiry CT/IA can help to externalise, compare and contrast participants' perspectives in a given context and facilitate self-discovery and establish more versatile styles of cognition and learning.

Diagrams or entailment meshes showing the dependencies of participant perspectives on concepts can assist this process. These n-ary relations can unfold from a head node, topic or concept and in IA or dynamic Lp, they can depict forces acting between concepts through PLT.

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